Checking positive definiteness or stability of symmetric interval matrices is NP-hard

summary:It is proved that checking positive definiteness, stability or nonsingularity of all [symmetric] matrices contained in a symmetric interval matrix is NP-hard

Complexity theoretic aspects of problems in control theory

Caption title.Includes bibliographical references (p. 5-6).Supported by the ARO. DAAL03-92-G-0115John N. Tsitsiklis

An interval-matrix branch-and-bound algorithm for bounding eigenvalues

We present and explore the behaviour of a branch-and-bound algorithm for calculating valid bounds on the k-th largest eigenvalue of a symmetric interval matrix. Branching on the interval elements of the matrix takes place in conjunction with the application of Rohn’s method (an interval extension of Weyl’s theorem) in order to obtain valid outer bounds on the eigenvalues. Inner bounds are obtained with the use of two local search methods. The algorithm has the theoretical property that it provides bounds to any arbitrary precision > 0 (assuming infinite precision arithmetic) within finite time. In contrast with existing methods, bounds for each individual eigenvalue can be obtained even if its range overlaps with the ranges of other eigenvalues. Performance analysis is carried out through nine examples. In the first example, a comparison of the efficiency of the two local search methods is reported using 4,000 randomly generated matrices. The eigenvalue bounding algorithm is then applied to five randomly generated matrices with overlapping eigenvalue ranges. Valid and sharp bounds are indeed identified given a sufficient number of iterations. Furthermore, most of the range reduction takes place in the first few steps of the algorithm so that significant benefits can be derived without full convergence. Finally, in the last three examples, the potential of the algorithm for use in algorithms to identify index-1 saddle points of nonlinear functions is demonstrated

NP-hardness of some linear control design problems

Includes bibliographical references (p. 12).Supported by the AFOSR. AFOSR-91-0368 Supported by the ARO. DAAL03-92-G0309 Supported by KTH, Stockholm.Vincent Blondel, John N. Tsitsiklis

A characterization of convex cones of matrices with constant regular inertia

AbstractLet A be a convex cone of n×n matrices. In this paper, we present a necessary and sufficient condition for A to contain matrices with a constant regular inertia, based on a version of the Lyapunov equation. The condition involves only the normalized extreme points of A. This extends a previous paper by the authors, where a robust stability criterion for A was obtained